Cobb-Douglas production is often given by $Y=AK^{\alpha}L^{1-\alpha}$, and $L$ is often just labor. But in RBC model, I notice that labor $h_t$ is restricted to $[0,1]$, and interpreted as working-time perecentage of non-sleeping time. Is this necessary restriction for RBC? Or can we allow $h_t$ to be in $[0,100]$ (with utility function $U(c_t,100-h_t)$)?

1 Answer
1

Specifications of this form treat labor $h_t$ and leisure $1-h_t$ as two alternative ways of spending a total time endowment. Labor can be used as an input to the production function, while leisure enters directly into the utility function as a desirable end in its own right.

In this environment, there are many equivalent normalizations. There is no need to say that the total time endowment is 1; instead, we could just as easily say it is 100, and by modifying the production function and utility function accordingly we'd get an isomorphic model. There's nothing special about any particular normalization.

That said, given a particular functional form for utility $u$, it often does matter how we define the time endowment, and this issue has played an important and somewhat embarrassing role in the history of RBC. In particular, if we normalize the total "time endowment" to 1 and say that $h_t$ is the labor fraction of the time endowment, then it can matter greatly whether we define the time endowment to be 24 hours every day for all individuals, limit to non-sleeping time, limit to non-sleeping time of working age adults, etc.

Why does this matter? The basic RBC model has typically assumed that intratemporal utility depends on a constant-returns-to-scale composite of consumption $c_t$ and leisure $l_t$. Given this, we can infer that this composite must have a constant elasticity of substitution of 1 from the fact that long-run labor supply seems roughly stable even as productivity and real wages have a strong upward trend. (See, for instance, pages 6 and 7 of Prescott 1986.) An elasticity of 1 means that the composite is Cobb-Douglas in consumption and leisure, taking the form $c_t^{1-\phi}l_t^\phi$ for some $\phi$.

RBC models then generally assume a constant intertemporal elasticity of substitution $1/\gamma$ for this composite, giving us the period utility function found on page 6 of Prescott (1986):
$$u(c_t,l_t) = \frac{[c_t^{1-\phi}l_t^\phi]^{1-\gamma}-1}{1-\gamma}$$
In the common special case with unitary intertemporal elasticity of substitution $1/\gamma=1$, the above reduces to simply
$$u(c_t,l_t) = (1-\phi)\log c_t + \phi \log l_t\tag{1}$$
Now, one important feature of a dynamic macroeconomic model is the Frisch elasticity of labor supply, which is the elasticity of labor supply to real wages holding marginal utility of consumption constant. This is important because it determines how the household will intertemporally substitute labor in response to the intertemporal pattern of real wages and real interest rates.

At an optimum the marginal utility of leisure must equal the real wage times the marginal utility of consumption, so for constant marginal utility of consumption $\lambda_t$ we have the condition
$$\frac{\phi}{l_t} = w_t\lambda_t$$
It follows that the Frisch elasticity of leisure $l_t$ with respect to $w_t$ is $-1$; this is a more-or-less direct consequence of the assumption that consumption and leisure form a Cobb-Douglas aggregate. But since $l_t = 1-h_t$, the local Frisch elasticity of labor $h_t$ is
$$\text{Frisch elasticity of labor}=\frac{dh_t/h_t}{dw_t/w_t}=\frac{-dl_t/h_t}{dw_t/w_t}\\
=-\frac{l_t}{h_t}\times \underbrace{\frac{dl_t/l_t}{dw_t/w_t}}_{\text{Frisch elasticity of leisure}} = \frac{1-h_t}{h_t}$$
From this, we can see that the definition of "time endowment" 1 matters greatly. For instance, from Table 1 of the most recent American Time Use Survey we see that average working hours per day are 3.14. If we take the time endowment to be all 24 hours, then $h=3.14/24 \approx .13$, and the implied local Frisch elasticity is about 6.6. If, on the other hand, we take out sleeping and other personal care activities, the remaining time endowment is 14.5 hours, in which case $h\approx.22$ and the implied Frisch elasticity is about 3.5.

Both these values are far higher than most Frisch elasticities estimated using micro data, as discussed by Chetty et al. (2012). Moreover, it seems bizarre that something as deep and important as the Frisch elasticity should depend on a choice as arbitrary as the definition of "time endowment". This connection is ultimately attributable to the assumption that $c_t$ and $l_t$ form a constant-returns-to-scale composite; King, Plosser, and Rebelo (1988) show that if we discard this assumption, a more general "KPR" functional form is consistent with the stylized fact of stable long-term labor supply, and we can potentially calibrate this more general form to the (much lower) elasticities estimated in the labor supply literature rather than imposing an unrealistically high Frisch elasticity by fiat.

For reasons that I do not completely understand, much of the RBC literature has followed Prescott (1986) and used the "log-log" utility parameterization (1) above, rather than calibrating the KPR functional form to match some externally estimated elasticity. This is probably due to a combination of inertia and convenience: the high Frisch elasticity guaranteed by the log-log parameterization helps to make labor volatile enough to match the macro data. And this is all connected to a lengthy and convoluted debate about the possible difference between "micro" and "macro" elasticities - see Chetty's slides or paper to read more about this.